cotangent bundle


Differential geometry

differential geometry

synthetic differential geometry








Given a manifold (or generalized smooth space) XX, the cotangent bundle T *(X)T^*(X) of XX is the vector bundle over XX dual to the tangent bundle T *(X)T_*(X) of XX. A cotangent vector or covector on XX is an element of T *(X)T^*(X). The cotangent space of XX at a point aa is the fiber T a *(X)T^*_a(X) of T *(X)T^*(X) over aa; it is a vector space. A covector field on XX is a section of T *(X)T^*(X). (More generally, a differential form on XX is a section of the exterior algebra of T *(X)T^*(X); a covector field is a differential 11-form.)

Given a covector ω\omega at aa and a tangent vector vv at aa, the pairing ω,v\langle{\omega,v}\rangle is a scalar (a real number, usually). This (with some details about linearity and universality) is basically what it means for T *(X)T^*(X) to be dual to T *(X)T_*(X). More globally, given a covector field ω\omega and a tangent vector field vv, the paring ω,v\langle{\omega,v}\rangle is a scalar function on XX.

Given a point aa in XX and a differentiable (real-valued) partial function ff defined near aa, the differential d af\mathrm{d}_a f of ff at aa is a covector on XX at aa; given a tangent vector vv at aa, the pairing is given by

d af,v=v[f], \langle{\mathrm{d}_a f, v}\rangle = v[f] ,

thinking of vv as a derivation on differentiable functions defined near aa. (It is really the germ at aa of ff that matters here.) More globally, given a differentiable function ff, the differential df\mathrm{d}f of ff is a covector field on XX; given a vector field vv, the pairing is given by

df,v=v[f], \langle{\mathrm{d}f, v}\rangle = v[f] ,

thinking of vv as a derivation on differentiable functions.

One can also define covectors at aa to be germs of differentiable functions at aa, modulo the equivalence relation that d af=d ag\mathrm{d}_a f = \mathrm{d}_a g if fgf - g is constant on some neighbourhood of aa. In general, a covector field won't be of the form df\mathrm{d}f, but it will be a sum of terms of the form hdfh \mathrm{d}f. More specifically, a covector field ω\omega on a coordinate patch can be written

ω= iω idx i \omega = \sum_i \omega_i\, \mathrm{d}x^i

in local coordinates (x 1,,x n)(x^1,\ldots,x^n). This fact can also be used as the basis of a definition of the cotangent bundle.

Revised on May 20, 2013 12:28:50 by Urs Schreiber (